The Mohr Circle for the passive case of a retaining wall carrying a sloping backfill is shown in FÃg.1. Consider an element subjected to a stress in the Ïƒv in the vertical direction and Ïƒx in the direction parallel to the backfill. As these stresses in one plane are parallel to the direction of another plane, these stresses are conjugate stresses and the planes are the conjugate planes.
The equivalent vertical stress-acting acting parallel to the surface of backfill is given by
Ïƒve = Ïƒvcosi = Î³zcosi
Referring to the figure the coefficient of passive earth pressure can be written as:
We can also write:
CD = OC sini
OD = OC cosi
R = OC sinÉ¸
By substitution these values in the expression for Kp we get:
Pp = KpÎ³z cosi
Kp = Kp cosi ———–(1)
And is known as the coefficient of passive earth pressure.
If, Î±1 and Î±2 are the inclination of failure planes with horizontal as shown in figure 2 then,
Ï´f2 = Î±1 – Ï´2
Î±1 = Ï´f2 + Ï´2
Î¨ = -i + 2Ï´2
Where, CD normal to OA
The pressure diagram is shown in figure 3 for both active as well as passive cases.